CHAPTER 1 Modeling the Vascular System and Its Capillary ......4 MODELING THE VASCULAR SYSTEM AND ITS CAPILLARY NETWORKS 1.3. THE VASCULAR SYSTEM It has long been known that the vascular

CHAPTER 1

Modeling the Vascular Systemand Its Capillary Networks

THOMAS H. DAWSON

United States Naval Academy, Annapolis, Maryland

Abstract. Modeling of the vascular system and its capillary networks is con-sidered within the context of associated scaling laws relating measurements ofcardiovascular variables of mammals ranging in size from the mouse to the humanand onward to the elephant. Topics include competing effects of viscous and iner-tial resistances to blood flow in the system and their appropriate separation formodeling purposes. Adequacy of the separation is confirmed by the demonstratedsuccess of the scaling theory in consolidating size-dependent measurements intoa single underlying description. Applications of the work include scaling of formand function for different-size systems directly to the human for confirmation ofphysiological descriptions and for practical matters such as the projection of safedrug dosage from small mammals to humans. Specific topics of form includediscussion of detailed scaling laws for radius and length of arterial and venousvessel and for the radius, length, and number of capillary vessels. Topics offunction include discussion of underlying physiological processes responsible forheart rate, ventricular outflow, blood circulation time, and kidney operation. Thescaling of therapeutic drug concentrations from mouse to human is consideredas an illustration of practical matters of interest.

1.1. INTRODUCTION

The vascular system of the body consists of the piping system of arteries and veinsand the various capillary networks that it serves. The piping system is responsi-ble for directing blood to the various capillary networks spread throughout thebody and for returning it to the heart for further circulation. The capillaries them-selves are responsible for transfer of life-sustaining substances between the bloodand surrounding tissues. When considering the subject in physiological studies,attention is usually focused on the vascular system of humans. However, there

Vascular Hemodynamics: Bioengineering and Clinical Perspectives, Edited by Peter J. YimCopyright 2008 John Wiley & Sons, Inc.

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COPYRIG

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2 MODELING THE VASCULAR SYSTEM AND ITS CAPILLARY NETWORKS

exists a strong similarity between the vascular system of the human and othermammals, and this similarity allows study of the system with greater perspectivethan would otherwise be possible.

This chapter is concerned with modeling the vascular system and its capil-lary networks for the purpose of interpreting related measurements on mammalsranging in size from the mouse to the elephant. With such modeling theory, dataon form and function for these different-size systems can be scaled directly to thehuman system for increased understanding and for application, as, for example,in drug therapy studies. The theory can also be used to establish important phys-ical aspects of the operation of the vascular system and its parent cardiovascularsystem through their dependence on scale. The work described here is based onearlier work of the author [1–4], although some new perspectives and results arepresented regarding theory and experimental support from existing measurements.

1.2. SOME BACKGROUND ON SCALING LAWS

Scaling laws for a physical system generally involve relations connecting mea-surable variables when all length measures of the system (e.g., lengths anddiameters of pipes in a piping system) are increased or decreased by the samefactor. Such scaling is referred to as uniform scaling . Relations for physiologicalvariables based on this concept were provided many years ago by Lambert andTeissier [5]. When applied to vascular hemodynamics, they are, however, inad-equate because of competing effects of inertial and viscous forces [6].

For perspective purposes, this inadequacy may be illustrated by consideringthe volume of ventricular blood VS forced into the vascular system during eachheartbeat, and its possible uniform scaling with mammal size. This quantity canbe expected to depend on the volume V of the ventricle, representative lengthsof the system l1 and l2 (among several), the contractile force F 0 of the ventricleper unit of wall area, a constant elastic modulus E with the same units as thecontractile force, the heart rate ω, and the viscous and inertial resistances to bloodflow as represented, respectively, by the coefficient of viscosity µ and density ρ

of the blood. Each of these variables has, of course, fundamental measuring units(or dimensions) associated with it. In the simplest cases, VS and V have units oflength cubed and l1 and l2 have units of length. Correspondingly, consistent withtheir definitions, F 0 and E both have units of force divided by length squared,ω has units of reciprocal time, µ has units of force times time divided by lengthsquared, and ρ has units of mass divided by length cubed.

Now, the governing equation connecting these variables must be expressiblein terms of dimensionless ratios in order to be independent of particular unitsemployed (meters or feet, newtons or pounds, etc.). With m-dimensional vari-ables and n fundamental units, there will, in fact, generally be m − n independentdimensionless ratios that can be formed. In the present case, there are nine vari-ables and four units involved: those of length, time, force, and mass. However,the units of force are also the units of mass times acceleration (mass times length

SOME BACKGROUND ON SCALING LAWS 3

divided by time squared) so that there are only three fundamental units involved.Accordingly, there are six independent dimensionless ratios to be considered. Aset of the required independent ratios can readily be formed by inspection, andthe governing relation for the problem thus written as

VS

V= f

(ρl2

1ω2

E,µω

E,F0

E,l2

l1,V

l31

)(1.1)

where f ( ) denotes an unspecified function.That each term in this equation is indeed dimensionless can be verified by

direct examination of the units involved. For example, the first term on the rightside can be written as ρl1 × l1ω

2/E , which has units of mass per length squared(ρl1) times acceleration (l1ω

2), divided by force per length squared (E ). Sincemass times acceleration has dimensions of force, the numerator has equivalentunits of force per length squared, like the denominator, so that all units canceland leave the ratio in the required dimensionless form.

As to physical interpretation, this first term can be regarded as the ratio ofinertial resistance to elastic restoring force of the ventricle and the second, as theratio of viscous resistance to this same elastic force. Similarly, the third term canbe regarded as the ratio of ventricular contractile force to the elastic restoringforce and the last two terms regarded simply as variables representing explicitdependence on size.

As to scaling requirements, the term on the left side of this equation representsthe ratio of two volumes and hence will be constant (or fixed) in uniform scalingfrom one mammal to another. The independent ratios on the right side must thenalso be fixed if scaling is to be possible. The last two terms in the equation areautomatically fixed because of the geometric similarity that exists with uniformscaling. The third from last term can be fixed when the contractile force F 0 perunit of ventricular wall area is scale-independent. However, a problem ariseswith the first two terms that are required to be fixed in uniform scaling. Forscale-invariant density and viscosity coefficient, as assumed here and as generallyconsidered appropriate [6], the first requires that heart rate vary inversely withthe length scale, while the second requires invariant heart rate among mammals.Thus, with these contradictory requirements, no scaling laws are possible underthe restriction of uniform scaling.

In the presently described work, the general requirement of uniform scalingis replaced with more restrictive conditions where the viscosity coefficient anddensity are combined appropriately with the linear dimensions of the separatevessels in which viscous and inertial effects dominate. This leads to non-uniformscaling laws for the length and radius of these vessels and to scaling laws forrelated physiological variables that are free of contradictions and consistent withmeasurements. In developing this subject, considerably more insight into vascu-lar hemodynamics is required than in the case of uniform scaling. Backgroundmaterial and related developments are thus considered here in some detail beforetaking up the development of the scaling laws.


1.3. THE VASCULAR SYSTEM

It has long been known that the vascular system consists of two main parts: thesystemic part, associated with transport of blood to the main body for exchange ofoxygen and other products; and the pulmonary part, associated with transport ofblood to the lungs for discharge of unwanted gaseous products and rechargeof oxygen. Discussion here will be directed mainly toward the systemic part ofthe circulation, but similar remarks will apply to the pulmonary part, and somereference will be made to it when possible.

A simplified representation of the various vessels of the systemic circulationis illustrated in Figure 1.1. Estimates of the size and number of these vesselsfor the complete systemic vascular system of the human are summarized inTable 1.1. These are projections of corresponding estimates for the dog by Green[7]. The scaling laws of the author [1] were used as a guide in making theprojections. The (unspecified) body mass of the dog was assumed equal to 14kg, based on Green’s values for the aortic radius (5 mm) and length (40 cm)and associated empirical relations of Holt et al. [8]. Some minor adjustmentsand rounding of values have been made in forming the estimates for the human(assumed body mass of 70 kg). For example, the aortic radius has been increasedby 20% to allow for vessel distension under mean blood pressure, and the radiusand length of the small arteries and veins have been chosen as the weightedaverages of the scaled values for the main and terminal branches of the originalwork. Noteworthy, for example, is the fact that the total volume of the vessels

Aorta

ArterioleCapillaries

Venule

Smallvein

Largevein

Vena cava Toheart

Fromheart

Largeartery

Smallartery

Figure 1.1. Branching of systemic vessels in mammals [1].

IDEALIZED VASCULAR HEMODYNAMICS 5

TABLE 1.1. Estimated Measures of Systemic Vascular Vessels of Humans

Vessel Radius (mm) Length (mm) Number Volume (mL)

Aorta 11 600 1 228Large arteries 3 300 40 339Small arteries 1 50 2400 377Arterioles 0.01 3 1.1 × 108 104Capillaries 0.005 1 3.3 × 109 259Venules 0.02 3 2.2 × 108 829Small veins 1 50 2400 377Large veins 5 300 40 943Venae cavae 11 500, 100 1 228

is equal to 3700 mL, which represents about 75% of the volume of the entire(systemic and pulmonary) vascular system and heart chambers and is consistentwith generally accepted estimates.

1.4. IDEALIZED VASCULAR HEMODYNAMICS

In addition to the size and number of the vessels, a further important characteri-zation can be made regarding resistance to blood flow during the cardiac cycle.This resistance may be divided into two parts: the inertial resistance associatedwith acceleration of the blood in the vessels and the viscous resistance associ-ated with velocity of the blood through the vessels. Of interest are the relativeimportance of these two resistances in the individual vessels and the generalmathematical forms that can be used to describe them. These matters will beinvestigated here using an idealized version of vascular hemodynamics based onaverage blood flow in the vessels at any instant rather than on the actual flow.Detailed variations of the blood flow in the arteries, arising from vessel disten-sion and resulting wave propagation, will thus be neglected, as will the irregularintermittent blood flow in the individual capillary networks. The adequacy ofresults from the simplified description will be checked initially in terms of bloodpressure predictions and then later in terms of scaling predictions derived usingthe description.

With the latter idealization, the inertial resistance f I (in units of pressure)provided by any one set of the parallel vessels listed in Table 1.1 may be estimatedusing the basic force–acceleration equation in the form

fI = βρL

πR2

Qr

N(1.2)

where ρ denotes mass density of the blood; L and R denote, respectively, thelength and radius of the individual vessel; N denotes the number of similar


parallel vessels; and Qr denotes the average blood acceleration (rate of changeof blood flow) in the set of vessels under consideration at any instant that resultsfrom the periodic cardiac contractions. Also, the coefficient β denotes a factor thattakes into account branching of vessels from the vessels under consideration. Forall except the nonbranching capillaries, its value may be estimated as one-half,corresponding to uniform branching with average flowrate equal to Qr/N at thevessel entrance and equal to zero at its far end. For the capillaries, the value ofthe coefficient is, of course, unity.

The viscous resistance fV (in units of pressure) for flow through any one set ofvessels may similarly be estimated using the standard Hagen–Poiseuille equationin the form

fV = β8 µL

πR4

Q

N(1.3)

where µ denotes viscosity coefficient of the blood and Q denotes the averageflow of the blood in the set of vessels at any instant. An implicit assumptioninvolved in the use of this equation is that sufficient time exists in the interval ofinterest to allow flow development from its initiation to fully viscous conditions.

In forming estimates with these equations, measured values of ventricularflowrate and flow will be used for the average flow conditions in the vessels.Well-known outflow characteristics of the left ventricle (measured in the ascend-ing aorta) are indicated in Figure 1.2 for humans, together with correspondingvalues of aortic blood pressure. Interest in the present work is in the initialstage (0.21–0.26 s) where extreme conditions exist, with an essentially constantflowrate of 11,000 mL/s2 and flow increasing from zero to a maximum of about550 mL/s.

The inertial and viscous resistances to blood flow at the ventricle arise from therespective sums of the inertial and viscous resistances of each class of vessels. Theresistive pressure at the far end of the system, where the blood empties into theright atrium, is essentially zero, so that the total resistance at the ventricle is thatinvolved in overcoming the flow resistances in the individual vessels. In applyingequations (1.2) and (1.3) for calculating these resistances, the mass density ofthe blood may be taken as 1.05 g/cm3 and the value for the viscosity coefficientas 0.03 dyn·s/cm2. The latter is appropriate for blood flow in macroscale arterialand venous vessels. However, within the small microscale vessels of the capillarynetworks, its value is less because of the nonhomogenous nature of the blood(plasma, components, and red blood cells) on this small scale. The actual value isunknown, and an approximate value of 0.02 dyn·s/cm2, about midway betweenthat of the plasma and the nominal value, will be used here.

Resulting calculations of the resistances for each vessel category are given inTable 1.2. A flow acceleration of 11,000 mL/s2 and a mean flow of 275 mL/shave been assumed, consistent with the early part of the ventricular data shownin Figure 1.2a. It can be seen that the inertial resistance in macroscale arterialvessels is essentially 100% of the total inertial resistance and that the viscousresistance in microscale vessels is essentially 100% of the total viscous resistance.

IDEALIZED VASCULAR HEMODYNAMICS 7

−100

0

100

200

300

400

500

600

700

800

0 0.2 0.4 0.6 0.8

Time (sec)

(a)

Ven

tric

ular

out

flow

(m

l/sec

)(0.26, 550)

Ventricular OutflowDuring Cardiac Cycle(Left Side of Heart)

(0.21, 0) (0.49, 0)

60

70

80

90

100

110

120

130

0 0.2 0.4 0.6 0.8

Time (sec)

Pre

ssur

e (m

m H

g)

Aortic ValveOpen

Aortic ValveClosed

Aortic Pressure

(b)

Figure 1.2. (a) Ventricular flow from left side of human heart and (b) correspondingblood pressures. (Data source: Selkurt and Bullard [9].)

Vessel dimensions alone limit the inertial resistance in the capillary networks,and calculated inertial resistances in the veins are ignored since measurementsindicate the absence of accelerated flow in these vessels (presumably because ofviscous damping in the capillary networks).

The total calculated resistance to the blood flow is 135,500 dyn/cm2 (102mmHg) as determined with the average flow of 275 mL/s. When the viscousresistance is greatest (for blood flow of 550 mL/s), the total calculated resistance


TABLE 1.2. Estimated Inertial and Viscous Resistances(dyn/cm2) in Systemic Vessels of Humans

Vessel Inertial Viscous Vessel Class

Aorta 91,200 430 MacroscaleLarge arteries 15,300 970 MacroscaleSmall arteries 380 220 MacroscaleArterioles 5.0 19,100 MicroscaleCapillaries 0.4 6,790 MicroscaleVenules 0.6 600 MicroscaleSmall veins — 220 MacroscaleLarge veins — 130 MacroscaleVenae cavae — 110 Macroscale

is 164,000 dyn/cm2 (123 mmHg) and when the viscous resistance is zero (forblood flow of zero) at the beginning of the ventricular contraction, the total resis-tance is 106,900 dyn/cm2 (80 mmHg). These calculated values are in excellentagreement with the known systolic (120 mmHg) and diastolic (80 mmHg) bloodpressures of healthy humans, as indicated in Figure 1.2b, and this suggests apossibly wider application of the simple theory and vessel characterization thanmight otherwise be expected.

An interesting consequence of this division between arterial and viscous resis-tances, perhaps worthy of note in passing, is that a reduction in the flow areaof the large arteries will increase the lower blood pressure and cause in turn themaximum and mean pressures to increase by the same amount. In contrast, areduction in the flow area of the arterioles will leave the lower blood pressureunchanged but will cause an increase in the maximum pressure equal to twicethe increase of the mean pressure.

1.5. IDEALIZED VESSEL CATEGORIES

The preceding results suggest for modeling purposes that the vascular systemmay be divided broadly into the categories of a connecting vessel between theleft side of the heart and the systemic capillary system where inertial resistanceis dominant, and the systemic capillary system itself, where viscous resistanceis dominant. In addition, it may also be considered to consist of a connectingvessel between the systemic capillaries and the right side of the heart, where vis-cous resistance also dominates. A similar idealized arrangement can be assumedapplicable to the pulmonary side, with a connecting vessel between the right sideof the heart and the pulmonary capillary system, and a connecting vessel fromthese capillaries to the left side of the heart. This idealization is illustrated inFigure 1.3.

THEORETICAL SCALING LAWS 9

Right heart

Left heart

Connectingvessels

Systemiccapillarysystem

Pulmonarycapillarysystem

Figure 1.3. Idealized description of the vascular system of mammals indicating collec-tive systemic and pulmonary capillary networks and connecting vessels [1].

A diagram like that in Figure 1.3 has often been used to illustrate the seriesconnection of the left and right sides of the heart in terms of pumping action[10]. In the present work, it has added meaning in illustrating the division ofthe flow characteristics of the blood, with inertial resistance dominating in thearterial connecting vessels from the left and right sides of the heart and viscousresistance dominating in the capillaries and in the venous connecting vessels tothe left and right sides.

1.6. THEORETICAL SCALING LAWS

Similar to earlier work of the writer, the conceptual model shown in Figure 1.3will be used to derive scaling laws that relate the dimensions of the arterialand venous vessels and the dimensions and number of vessels in the capillarynetworks for mammals of vastly different sizes [1]. The theoretical presentationhere will be a simplified treatment of the early work and will be based on thenecessary requirement that any physical equation be independent of particularunits employed and hence must be expressible in dimensionless form.

Resting conditions will be assumed as the basic conditions governing thesizing of the vascular system of different mammals, consistent with results fromearlier work. As in the previous discussion of idealized vascular hemodynamicsand associated blood pressure calculations, average flow in the blood vessels atany instant will be assumed to be representative of the actual complex flowsinsofar as scaling matters are concerned.

A typical dependent variable of the cardiovascular system, considered previ-ously in connection with equation (1.1), is the stroke volume of the left side ofthe heart, that is, the total volume of blood ejected into the systemic vascularsystem during each contraction of the left ventricle. In a more restrictive sensethan earlier, this can now be expected to depend on the volume of the ventricle


V , a typical length dimension l of the ventricle, the contractile force F 0 perunit area of the ventricular wall (measuring the strength of the contraction), ascale-independent constant E (in units of force per unit area) associated with thegeneral relation between fractional volume change of the ventricle and contrac-tile force in the absence of restraint from blood flow, the heart rate ω, and theinertial and viscous resistances to blood flow in the vascular system, as repre-sented, respectively, by equations (1.2) and (1.3), with constant blood density ρ

and constant viscosity coefficients µc and µv associated with blood flow in themicroscopic capillary vessels and macroscopic venous vessels, respectively.

Now, from equations (1.2) and (1.3), it can be seen that the expression forthe inertial resistance to blood flow has both the density of blood ρ and the ratioL/NR2 appearing together as a product and that for the viscous resistance hasboth the blood viscosity µ and the ratio L/NR4 appearing as a product. Thus, thestroke volume can be expected to depend not on the blood density and viscosityby themselves, but rather as they appear in these two products. Consideringindependent dimensionless ratios of the variables, the stroke volume VS mayaccordingly be expressed in general mathematical form as

VS

V= f

(ρLaω

2V

R2aE

,µcLcωV

NcR4cE

,µvLvωV

R4vE

,F0

E,V

l3

)(1.4)

where f ( ) denotes an undefined function of the indicated variables and Ra andLa denote the radius and length of the connecting arterial vessel, respectively; Rc,Lc, and N c denote the radius, length, and number of the characteristic capillaries;respectively; and Rv and Lv denote, respectively, the radius and length of thevenous connecting vessel. Consistent with the earlier discussion, single arterialand venous connecting vessels have been assumed so that N a and N v are bothequal to 1 in the conceptual model.

The physical interpretation of the terms in this equation is similar to thatdiscussed earlier in connection with equation (1.1). The first term representsthe ratio of inertial resistance in the arteries to the elastic ventricular restoringforce, and the second and third terms represent the ratios of viscous forces in thecapillaries and veins to this same force. Similarly, the fourth term represents theratio of ventricular force to elastic restoring force, and the last term representsthe ratio of ventricular volume to the cube of any of its linear length measures.As will be seen below, the explicit inclusion of separate measures for the arteries,capillaries, and veins in this equation now allow scaling relations to be developedwithout the contradictions associated with equation (1.1).

Measurements of Holt et al. [11] have shown that the ratio of stroke volumeto ventricular volume, denoted by the left side of equation (1.4), is constantfor mammals ranging in size from the rat to the horse, consistent with the ideaof similitude in mechanical systems having different sizes. This condition isillustrated in Figure 1.4 for typical mammals. The implication is that the dimen-sionless variables on the right side of equation (1.4) are also constant underchange of scale. These ratios provide the conditions needed for similarity andscale modeling of the system.


0

0.2

0.4

0.6

0.8

0.1 1 10 100 1000 10000

Ventricular Volume (ml)

Str

oke

/ Ven

tric

ular

Vol

ume

Holt et al. [11]

Average

Rat

Horse

Ratio of Stroke Volumeto

Ventricular Volume

RabbitDog Sheep

Figure 1.4. Measurements (circles) showing that the ratio of stroke volume to ventriclevolume is essentially constant for mammals ranging from rats to horses.

The ventricular volume V represented in equation (1.4) is known from thework of Holt et al. [11] to vary directly with mammal mass M under change ofscale so that the condition for the first three terms on the right side to remainfixed for different mammals may be written as

Laω2

R2a

∝ M−1,Lcω

NcR4c

∝ M−1,Lvω

R4v

∝ M−1 (1.5)

where the symbol ∝ denotes proportionality under change of scale and wherethe density and viscosity coefficient of the blood are assumed scale-independent.The fourth term in equation (1.4) requires also that the contractile force F 0

per unit area of ventricular wall remain fixed under scale change, and the fifthterm requires that all important length dimensions of the ventricle vary in thesame way, as ventricular volume raised to the one-third power. Since ventricularvolume varies directly with mammal mass, the last condition also means that theselength dimensions must vary as mammal mass raised to the one-third power.

In addition to relations (1.5), three other relations may be written associatedwith the fact that the total blood volume in mammals is known to vary directlywith mammal mass [12]. As parts of the whole, the blood volumes in the connect-ing vessels and the capillary system can therefore similarly be considered to varyin this manner. The following additional relations represent this requirement:

R2aLa ∝ M, NcR

2cLc ∝ M, R2

vLv ∝ M (1.6)

Relations (1.5) and (1.6) provide six relations between the eight variablesinvolved, that is, the seven variables associated directly with the vascular system


and the heart rate. Two additional relations are thus needed for determining theirvariations with mammal mass. A theoretical approach involving basic cardiaccell processes was taken in the first work of the author [1] in order to obtainthese needed relations and an empirical one in later works [2,3]. Here, attentionwill be directed again to the theoretical approach, with perhaps some furtherclarification.

The main idea behind the theoretical development of the needed additional rela-tions is that the scaling relations for the variables associated with the characteristiccapillary system can be expected to apply (in the resting state) to the capillaries ofany organ of the body. Thus, the number of capillaries in the heart can be consideredto be proportional under scale change to the number of capillaries N c referred to inrelations (1.5) and (1.6). The number of cardiac cells in the heart can also be con-sidered to be proportional to the number of capillaries supplying them. Thus, tissuevolume of the heart can be considered proportional under scale change to the prod-uct of capillary number and the volume of a single cell. Since tissue volume of theheart and heart mass are proportional, the latter can also be substituted for the formerin this last relation. Moreover, heart mass and body mass are known to be propor-tional [12] under scale change, so that, on inverting the tissue–volume relation andusing this last condition, the volume of a single cardiac cell is predicted from thesearguments to be proportional to the ratio M /N c.

This last relation provides the connection between cardiac-cell size and vari-ables already present in the discussion, and this in turn allows further descriptionin terms of fundamental processes. For this purpose, it is convenient to use a sim-ple length measure of a cell. Assuming all lengths of the cells scale in the sameway, the cell volume must then be proportional to the cube of any length measure,and a characteristic length d of a cell can therefore be expressed simply as

d ∝(

M

Nc

)1/3

(1.7)

Insofar as application is concerned, this last relation has significance becausecardiac tissue is known to consist mainly of cardiac fibers which, when excited,contract and provide the pumping action of the heart. These fibers are alsoknown to consist of series connections of cardiac cells. The diameter of thecells and hence that of the fibers can thus be expected to be proportional underchange of scale to the characteristic dimension d defined by relation (1.7).

With respect to operation, it is known that resting cardiac fibers, like allmuscle fiber, have an excess of positive ions immediately outside, relative totheir interiors. Contraction is initiated in the upper heart by the relatively slowmovement of positive ions into specialized fibers until the inside potential israised to a critical value needed for rapid influx of additional ions under theelectrical attraction existing. This sudden increase generates a voltage pulse (oraction potential), which then propagates across the heart, causing contraction andpumping action. At the same time, the newly arrived positive ions in the fibersare forced out by a metabolic energy source and the potential at the initiation site


is returned to its initial value. The process repeats itself periodically to providethe heartbeat and cyclic pumping action.

On the basis of this brief description, two issues can be considered for obtain-ing the additional relations needed to complete the scaling description of thevascular system: (1) the initial cyclic slow influx of ions into the fibers at theinitiation region of the heart and (2) propagation of the contraction signal overthe heart fibers to form the full heartbeat.

The first issue can be addressed by likening the slow influx of ions to a generaldiffusion process. The relation for the mass m of an ionic substance moving intothe cells of a fiber is then expressible in dimensionless terms as

m

�Cd3= f

(ωd2

D

)(1.8)

where f ( ) denotes a general function, D denotes a constant (in units of areaper time), and �C denotes the initial difference between concentrations outsideand inside the cell (in units of mass per volume). The left side of this equation,with concentration difference �C assumed scale-invariant, is proportional to themass influx per unit cell volume, and this may be expected to be independentof mammal size. With the constant D also scale-invariant, as expected, it can beseen that the product of heart rate and square of cell dimension must likewise beconstant under change of scale:

ωd2 ∝ M0 (1.9)

Using the definition of cardiac cell dimension of relation (1.7), the follow relationthus results

ω ∝ M−2/3N2/3c (1.10)

For the second matter, it is assumed that the propagation speed of the voltagepulse through cardiac fiber is, like nerve fiber, expressible as a power-law relationwith fiber diameter; that is, as db, where b denotes a constant. From experimentalstudies with small isolated nerve fibers (generally considered analogous to heartfibers), a value of b can be expected to be between 0.57 [13] and 0.78 [14].A value of b equal to 2

3 was used by the author in earlier work [1] and willbe assumed here. Now, the period between heartbeats (reciprocal of the heartrate) must be proportional to the ratio of heart length traveled by the voltagepulse to the propagation speed. Assuming that overall heart dimensions scalewith mammal mass to the one-third power (like ventricular dimensions notedearlier), the following relation can be obtained:

ω ∝ M−1/9N−2/9c (1.11)

Relations (1.10) and (1.11) provide the scaling laws for the heart rate and capillarynumber, and the second of relations (1.5) and (1.6) then provide the scaling laws


for the radius and length of capillary vessels. The results are expressible in theform

ω ∝ M−1/4 (1.12)

Rc ∝ M1/12, Lc ∝ M5/24, Nc ∝ M5/8 (1.13)

It is worthwhile to note that this solution for the scaling laws of the capillaryvessels is independent of the assumption that inertial resistance dominates in themacroscopic arterial system and depends on the less stringent assumption thatviscous effects dominate in the microscopic vessels, irrespective of the conditionsapplying in the macroscopic system.

With the above solution available, the first and third of relations (1.5) and(1.6) then provide the remaining scaling laws for the radius and length of themacroscale arterial and venous vessels. The solutions are expressible as

Ra ∝ M3/8, La ∝ M1/4, Na ∝ M0 (1.14)

and

Rv ∝ M7/24, Lv ∝ M5/12, Nv ∝ M0 (1.15)

where it is also noted that the number of arterial and venous connecting vesselsNa and Nv are invariant with scale change.

1.7. OBSERVATIONS ON THE SCALING LAWS

Heart Rate

The reciprocal one-fourth scaling law for heart rate of relation (1.12) is intimatelyconnected to the scaling laws for the vascular system, and its validity is thereforeessential to the validity of the scaling laws themselves. The law is in fact consis-tent with the generally accepted empirical relation for resting mammals that hasexisted for many years [11,12,15,16]. Typical data are indicated in Figure 1.5.Predictions are also shown from the scaling law when expressed with averagecoefficient as

ω = 230M−1/4 (1.16)

The agreement between theory and measurement can be seen to be very goodover the entire range from mouse to elephant. When applied to the human havingtypical body mass of 70 kg, the equation predicts a heart rate of 80 beats/min,which is about 14% greater than the value of 70 beats/min that usually appliesfor resting and relaxed conditions. For predictions for such a state, the coefficientshould therefore be reduced from 230 to 200.

OBSERVATIONS ON THE SCALING LAWS 15

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2 2.5 3

Mammal Mass (kg) to minus 1/4 power

Hea

rt R

ate

(bea

ts /

min

)

Clark [15]

Holt et al. [11]

230 M−1/4

Heart Rate Mouse

Guinea Pig

Elephant[12]

RatGoat

Rat [16]

Rabbit

Figure 1.5. Measurements (symbols) confirming the scaling law for resting heart rate,with data from typical mammals identified.

It is interesting to note that the physical explanation for increased or decreasedheart rate of mammals relative to the human is provided by relation (1.9), whichrequires that heart rate vary inversely with the square of cardiac cell dimensions.Thus, for example, the diameter of the cardiac cells of the mouse must (accordingto this description) be approximately one-third smaller than that of the human inorder to provide the observed increase in the resting heart rate by a factor of 9.

Ventricular Outflow

As a further preliminary examination of the vascular scaling theory, the relatedscaling laws associated with ventricular outflow will be developed and comparedwith measurements. These relations will be needed in the following work wherethe applicability of the theory is examined for mammals larger and smaller thanthe human (in the manner of results already presented in Table 1.2 for the human).For this purpose, it is convenient to return to the basic relation (1.4) and replacethe dependent dimensionless stroke volume on the left side with the dimensionlessratio of flow Q to the product of ventricular volume V and heart rate ω, andalso include the dimensionless time ωt on the right side. Similarity requires thatthis variable and all the original variables in the relation to remain fixed underchange of scale. Ventricular volume is proportional to mammal mass under scalechange, as noted earlier, so that flow in this revised relation at any fixed timeωt is predicted to vary directly as the product of mammal mass and heart rate.


A corresponding treatment indicates that flowrate Qr must vary directly with theproduct of mammal mass and the square of heart rate at any fixed time ωt . Thus,the scaling relations for ventricular flow and flowrate at any time t in the processmay thus be written as

Q ∝ M3/4, Qr ∝ M1/2, t ∝ M1/4 (1.17)

With these relations, the entire time history of ventricular outflow of one mam-mal can be predicted from that of another. An illustration of their applicability isshown in Figure 1.6, where the outflow history for the human, as indicated ear-lier in Figure 1.2, has been scaled to the horse and compared with measurementsreported by Noordergraaf et al. [17]. Here, a typical body mass of 70 kg wasassumed for the human and an effective mass of 450 kg for the horse. No specificvalue was given for the latter in the cited reference, and the effective value wasdetermined using the known condition that the ratio of areas under the two flowhistory graphs is equal to the ratio of stroke volumes and hence is equal to the cor-responding ratio of mammal masses. The agreement shown in Figure 1.6 can beseen to be remarkably good considering the difference in scale between the two.

Range of Applicability

The general applicability of the scaling laws for the macroscale arterial vesselsof the vascular system depends on the continued dominance of inertial resistance

−400

0

400

800

1200

1600

2000

2400

2800

0 0.2 0.4 0.6 0.8 1 1.2

Time (sec)

Ven

tric

ular

out

flow

(m

l/sec

)

Horse (450 kg) Scaled from

Human

Ventricular OutflowDuring Cardiac Cycle (Left Side of Heart)

(0.78, 0)

Human(70 kg)

(0.33, 0)

Data fromNoordergraaf

et al. [17]

Figure 1.6. Ventricular outflow of horse as scaled from the human and as reported(circles) by Noordergraaf et al. [17].

OBSERVATIONS ON THE SCALING LAWS 17

to blood flow in these vessels as attention is shifted to mammals larger andsmaller than the human. This matter can be investigated by projecting vesselcharacteristics of the human (Table 1.1) to the horse and rat using the previouslydeveloped vascular scaling laws and then calculating the flow resistances, asdone earlier for the results given in Table 1.2, with the help now of the scalingrelations (1.17).

Calculations of the vessel resistances for the horse (of 450 kg body mass) arelisted in Table 1.3 for scaled values of flow acceleration and one-half maximumflow (27,900 mL/s2, 1110 mL/s at 0.374 s). When compared with those given inTable 1.2, it can be seen that a scale change from the human to the horse leavesunchanged the inertial resistances in the arteries and the viscous resistances in thecapillary networks and veins. This is as expected from the theory. The viscousresistances in the arteries do change some but are still negligible overall comparedwith the inertial resistances and are, in fact, somewhat less than those in thehuman because of the increase in scale. Calculated blood pressures accordinglyremain unchanged from those determined for the human. The scaling relationsderived under the assumptions of dominant inertial resistances in the arteries canthus generally be expected to apply at least to mammals of the size of the humanand larger.

Similar calculations are also shown in Table 1.3 for the rat (0.3 kg body mass),using scaled values of flow acceleration and one-half maximum flow (720 mL/s2,4.6 mL/s at 0.060 s). The inertial resistance in the arteries and the viscous resis-tances in the capillary networks and veins are found to remain unchanged, asin the case of horses and humans. However, the total calculated viscous resis-tance in the arteries is found to be about 23% of the total inertial resistanceand is therefore no longer negligible. The calculated diastolic blood pressureremains the same as for the human and the horse because this depends only onthe inertial resistance in the arteries, which continues to be scale-independent.The calculated systolic blood pressure (for maximum flow) does change, though,

TABLE 1.3. Estimated Flow Resistances (dyn/cm2) forHorse and Rat at One-Half Maximum Flow

Horse (450 kg) Rat (0.3 kg)

Vessel Inertial Viscous Inertial Viscous

Aorta 91,200 170 91,200 6,580Large arteries 15,300 380 15,300 14,900Small arteries 380 90 380 3,340Arterioles 4 19,100 0.8 19,100Capillaries 0.4 6,790 0.7 6,790Venues 0.5 600 1.0 600Small veins — 220 — 220Large veins — 130 — 130Venae cavae — 110 — 110


from 122 to 158 mmHg, and this is a direct result of the appreciable viscousresistance calculated for the arteries of the rat. These calculations suggest thatrelations (1.14) for scaling the macroscale arterial vessels, and the associatedprediction of identical blood pressures, could not apply for smaller mammalssuch as the rat.

The latter observation is, however, inconsistent with the original blood pressuremeasurements of small mammals by Greg et al. [18] and Woodbury and Hamilton[19], which showed values for the mouse and dog to be the same as those forthe human. More recently, Marque et al. [16] have reported measurements inconnection with other work that confirm these earlier results specifically for therat. The indication is therefore that viscous resistances in the arteries are, in fact,much less than calculated values from equation (1.3). In contrast, the calculationsin Table 1.3, as well as those in Table 1.2, indicate that equation (1.3) does indeedapply to the capillary networks.

Evidence for the legitimacy of both of these indications can be found in thetheory of starting motion of viscous fluids under constant driving pressure, asdeveloped by Szymanski [20]. According to this theory, the initiated flow willfirst resemble that occurring in the absence of viscous effects (Figure 1.7a) andthen proceed to become the fully developed Hagen–Poiseuille flow (Figure 1.7b),on which equation (1.3) is based (with the constant driving pressure then equalto the viscous resistance for the resulting flow). For fully developed flow in avessel of radius R after time τ since initiation, the theory requires the value ofthe ratio ρR2/µτ to be equal to or less than unity, with ρ and µ denoting blooddensity and viscosity coefficient, respectively, as earlier.

If the driving pressure continually increases during the period τ (as in ven-tricle outflow during the period of interest here), smaller time intervals must be

(a) (b)

Rat (0.3 kg) with t = 0.013 s

Vessel Radius (cm) rR2/mt ConditionAorta 0.14 53 UnderdevelopedLarge artery 0.04 4 UnderdevelopedSmall artery 0.01 0.3 UnderdevelopedCapillary 0.0003 0.0004 Fully developed

Figure 1.7. Velocity distributions illustrating flow development in blood vessels fromstarting condition [underdeveloped, early flow (a)] to fully developed viscous flow [laterflow (b)] and illustration of its measure with the dimensionless number indicated.

EXAMINATION OF SCALING LAWS FOR VESSEL DIMENSIONS 19

considered for which the condition of constant driving pressure is effectivelysatisfied. Thus, taking the smaller increments as τ /100, the criterion for fullydeveloped flow in each interval is expressible as ρR2/µτ ≤ 0.01. Application ofthis requirement for the rat is included in the flow illustrations in Figure 1.7, withthe time τ measured from the beginning of the heart cycle and equal to that atwhich maximum flow occurs.

It can be seen from the results in Figure 1.7 that the values of the flowdevelopment ratio for the arterial vessels of the rat are not small in the requiredsense for fully developed flow, in contrast with the capillary vessel, where thiscondition is satisfied to an extent far exceeding minimum requirements. Theconclusion is therefore that viscous flow is not fully developed in the arterialvessels and that equation (1.3) cannot be expected to apply to flow resistances inthese vessels, but that this is not the case for the vessels of the capillary networks.

The matter of apparent viscous resistance in the arterial vessels of the rat andother small mammals is thus clarified; calculations with equation (1.3) greatlyoverestimate the actual viscous resistance in the arteries so that, as with largermammals, inertial resistance dominates in the arterial vessels and viscous resis-tance dominates in the capillary networks and the venous vessels. The vascularscaling relations developed under this condition are thus applicable to smallermammals such as the rat as well as to larger ones such as the horse. Blood pres-sures in these circumstances are also predicted to be the same for both small andlarge mammals.

Within the capillary networks, the scaling laws of relations (1.13) can beexpected to apply to the number and dimensions of the capillaries for all mam-mals. These relations depend only on the assumptions leading to relations (1.10)and (1.11) and the second of relations (1.5) and (1.6). In fact, they may alsobe established using the requirement that capillary blood volume is proportionalto mammal mass, together with the relation for scale-independent drop in bloodpressure across the capillary networks and the scaling relation for oxygen trans-fer to surrounding tissue [3]. The scaling relations (1.13) will also apply to themicroscopic arterioles and venules that connect to the capillaries, as assumedhere, provided only that their numbers scale directly with the capillary numbers.

The scaling laws of relations (1.15) for macroscale venous vessels can sim-ilarly be expected to apply to the upper and lower vena cava and to the largeveins of all mammals. They may also apply to the small veins, as assumed here,provided their number is scale-independent like that of the larger veins.

1.8. EXAMINATION OF SCALING LAWS FOR VESSEL DIMENSIONS

The scaling laws for vessel dimensions are compared below with measurementsreported by others for arterial, capillary, and venous vessels. In each case con-sidered, the scaling relation has been cast in equation form, with its coefficientdetermined from a best-fit (least-square) analysis of the data involved.


Arterial Vessels

Holt et al. [8] measured the aortic radius (in the absence of any significant vesseldistension from mean blood pressure) over the range from mouse to cow. Similarautopsy measurements of Clark [15] of the aortic radius are also available. Thefirst of relations (1.14) requires the aortic radius to vary with mammal massto the power 3

8 . Figure 1.8 illustrates the excellent agreement existing betweenthe measurements (ascending aorta) and theoretical predictions when the scalingrelation is expressed in the form indicated in the figure. Similar agreement existswith the earlier data of Clark.

Holt et al. [8] also measured the length of the aorta (heart to bifurcation)for the same mammals as used for the aortic radius. These data are shown inFigure 1.9 together with predictions from the second of relations (1.14) in theform of the equation indicated.

The theoretical scaling predictions for the aortic length can be seen to agreevery well with the measurements from mammals ranging from the rabbit to thecow. There is, however, some overestimate of the measured values from mouseand rat. For these mammals, it may simply be that other conditions enter toprevent closer agreement with the theoretical projections.

Capillary Vessels

Measurements of capillary vessels are limited for the systemic side of the cir-culation, but the data available are consistent with the present theory [1,3].Fortunately, for the pulmonary side of the circulation there are estimates (based

0

0.5

1

1.5

2

2.5

0 2 4 6 8 10 12 14

Mammal Mass (kg) to Power 3/8

Aor

tic R

adiu

s (c

m)

RA = 0.18 M 3/8

Data ofHolt et al. [8]

Aortic Radius

Mouse

Horse Cow

Goat

Rat

RabbitDog

Figure 1.8. Comparison of data with 38 th scaling law for aortic radius.

EXAMINATION OF SCALING LAWS FOR VESSEL DIMENSIONS 21

0

20

40

60

80

100

120

0 1 2 3 4 5 6


Aor

tic L

engt

h (c

m)

Mouse

Data of Holt et al. [8]

LA = 21 M 1/4

Aortic Length

Rat

Cow

Dog

Rabbit

Goat

Horse

Figure 1.9. Comparison of data with 1/4-th scaling relation for aortic length.

on tissue samplings) of the net volume and surface area of the capillaries presentin the oxygen transfer surfaces of the lung of mammals [21]. These data havebeen examined for individual mammals and the capillary radius and net capil-lary length determined from simple geometry. The results have been consideredearlier [1,2]. Typical values for the capillary radius are shown in Figure 1.10together with predictions from the first of relations (1.13) in the form of the 1

12 thpower law indicated.

It can be seen that considerable scatter exists in the data, so that nothingdefinitive can be said about the predictions other than that they follow the trend ofthe data and provide a good representation of their average variation. The data doconfirm the theoretical indication of a dependence of capillary radius on mammalsize, and this alone provides an endorsement of the theoretical description. It alsooffers a striking illustration of the level of detail present in the vascular systemof all mammals.

Similar data are shown in Figure 1.11 for the net length of the pulmonarycapillaries, defined as the product of the capillary length and the number present.From the second and third of relations (1.13), the scaling relation for this netlength involves mammal mass to the power 5

6 . Predictions from the indicatedequation having this form are also shown in the figure. Here the scatter is minimaland the data and predictions can be seen to agree very well with one another.

Venous Vessels

Considering next the venous vessels, the measurements of Holt et al. [8] mayagain be used. Figure 1.12 shows these data for the radius of the inferior vena


0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0 0.5 1 1.5 2 2.5


Cap

illar

y R

adiu

s (m

m)

RC = 0.0027 M 1/12

Data based onGehr et al.

[21]

Pulmonary Capillaries Radius

Shrew

Cow

DogMouse

Figure 1.10. Data from pulmonary side of circulation for capillary radius compared with1

12 th relation.

0.1

1

10

100

1000

10000

100000

0.001 0.01 0.1 1 10 100 1000


Net

Cap

illar

y Le

ngth

(km

)

NCLC = 160 M 5/6

Shrew

Pulmonary Capillaries Net Length NCLC

Cow

Dog

MouseData based on

Gehr et al.[21]

Figure 1.11. Data from pulmonary side of circulation for net capillary length comparedwith 5

6 th relation.

APPLICATIONS OF SCALING THEORY 23

0

0.5

1

1.5

2

2.5

3

0 1 2 3 4 5 6 7


Ven

a C

ava

Rad

ius

(cm

)Inferior Vena Cava

Radius

Data ofHolt et al. [8]

RV = 0.38 M 7/24

Rat

Rabbit

Goat

Dog

Dog

Cow

Horse

Figure 1.12. Comparison of data with 724 th scaling law for radius of vena cava.

cava. Predictions are also shown from the first of relations (1.15) when expressedin the form indicated. Some variability can be seen to exist in the measurements,and this renders the comparisons less definitive than might otherwise have beenthe case. Nevertheless, the theoretical predictions do provide a good averagerepresentation of the data.

Measurements of the length of the vena cava for various mammals are shownin Figure 1.13. In analyzing the data, the measurement for the horse appearedto be an anomaly and was neglected in determining the best-fit value for thecoefficient of the scaling law. The remaining measurements can be seen to agreewell with the predictions from the theoretical scaling relation.

1.9. APPLICATIONS OF SCALING THEORY

Indirect applications of the scaling laws for the vascular system have alreadybeen considered for heart rate, ventricular outflow, and blood pressures. Moredirect applications are described below concerning cardiac output and oxygendelivery, capillary density, circulation time of the blood, and kidney form andfunction. A final illustration involves the scaling of the time history of therapeuticdrug concentration in the blood from the mouse to the human.

Cardiac Output and Oxygen Delivery

As is well known, a major role of the vascular system is to direct blood from theheart to the systemic capillaries for oxygen transfer to the cells. The transport


0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14


Ven

a C

ava

Leng

th (

cm)

Inferior Vena Cava Length

Data of Holt et al.[8]

LV = 9.3 M 5/12

Rat

Rabbit

Dog

Goat

Cow

Horse

(?)

Figure 1.13. Comparison of data with 512 th scaling law for length of vena cava.

activity is conveniently characterized by the cardiac output, which denotes theflow of blood into the system when averaged over a heart cycle. Cardiac output isthus equal to the product of stroke volume and heart rate and scales as mammalmass to the power 3

4 , consistent with the measurements shown in Figures 1.4and 1.5.

For the resting mammal, mixed venous blood is approximately 75% saturatedwith oxygen (with maximum of 20 mL per 100 mL of blood), so that about5 mL of oxygen is delivered to the cells for every 100 mL of blood from theheart. Oxygen consumption rate (or metabolic rate) is thus equal to 5% of thecardiac output and accordingly scales also as mammal mass to the power 3

4 .This scaling relation has been known for many years to be a result of directmeasurements of oxygen utilization of mammals.

The associated rate of transfer of oxygen from the capillaries to surround tissueinvolves a basic diffusion process with driving force equal to the difference inoxygen partial pressures �PO inside and outside the capillaries. The transfer rateis also proportional to the net surface area of the capillaries NC × (2πRCLC) andinversely proportional to their wall thickness HC. Thus, oxygen transfer rate VO2

is expressible as

VO2 ∝ �PONCRCLC

HC

(1.18)

Assuming that the partial pressures inside and outside are proportional underscale change and that the wall thickness and radius are similarly proportional,


the following simplified relation results

VO2 ∝ PONCLC ∝ M3/4 (1.19)

where PO denotes oxygen pressure in the blood and the second proportionalitydenotes the required scaling dependence of oxygen consumption rate noted above.The product NCLC scales as mammal mass to the power 5

6 so that the oxygenpressure in the blood must scale as mammal mass to the power– 1

12 if the relationis to give the required 3

4 th scaling with body mass. This has, in fact, been shownto be the case by the author [1], based on detailed analysis of measurements ofSchmidt-Nielsen and Larimer [22] for a saturation level of 75% or so.

Capillary Density

The number of capillaries per unit cross-sectional area of tissue is referred to ascapillary density . It generally increases with decreasing mammal size, at least inmuscle tissue, where the parallel arrangement of capillary vessels makes system-atic investigation possible. Because of this arrangement, a simple scaling relationfor capillary density of muscle can be established. In particular, the characteristicnumber of capillaries NC per unit of cross-sectional area perpendicular to thecapillary lengths must be proportional to the product of the volume density ofthe capillaries (number per unit of associated muscle volume) and the character-istic capillary length. With volume density proportional to muscle mass densityand with muscle mass proportional to mammal mass, as is known to be the case,capillary density γ is thus expressible as

γ ∝ NC

M× LC or γ ∝ M−1/6 (1.20)

where the scaling relations (1.13) have been used in writing the second expres-sion.

The variation of capillary density with mammal size has been studied indetail [23], and data for the masseter (jaw) muscle of mammals are shown inFigure 1.14. Also shown are predictions from the scaling relation (1.20) whencast in the indicated equation form.

Although some variability exists in the data, the scaling predictions can beseen to provide a good representation of the average increase associated withdecreasing mammal size. Interestingly, the scaling relation (1.20) can also beestablished by assuming that capillary spacing is proportional to capillary radius.This, in turn, indicates geometric similarity under scale change in the array ofcapillaries and spacings in the cross-sectional areas of the muscle.

Circulation Time

As might be anticipated, the average time required for complete circulation of asmall volume of blood around the vascular system is dependent on mammal size.


0

1000

2000

3000

4000

0 0.5 1 1.5 2 2.5

Mammal Mass (kg) to the Minus 1/6 th Power

Den

sity

(nu

mbe

r pe

r m

m2 )

Capillary Density (Masseter Muscle)

Data of Schmidt- Nielsen and

Pennycuik [23]

g = 1300 M −1/6

Bat

Pig

Mouse

Rat

Guinea Pig

Cat

Rabbit

Dog

Figure 1.14. Comparison of measurements with theory for capillary density.

This matter is readily discussed with the help of the present theory. Referringin particular to the idealized description illustrated in Figure 1.3, the circulationtime for the systemic system involves the time for travel through the arterialconnecting vessel, the capillary network, and the venous connecting vessel. Thetime of travel through each of these vessels is proportional to the length of thevessel L divided by the average velocity of the blood through it.

Considering, for example, the capillary network, this velocity is equal to thecardiac output Qb divided by the total cross-sectional area of the network so thatit is proportional to the ratio Qb/NCRC

2. On substituting the scaling relations forthe capillary vessels and the previously discussed 3

4 th relation for cardiac output,the velocity of the blood in the capillaries is found to be proportional to mammalmass to the power − 1

24 . Capillary length scales as mammal mass to the power5

24 , so the travel time is thus found to be proportional to mammal mass to thepower 1

4 . The same can be found true for the connecting vessels. Thus, the entiretravel time for an element of blood through the systemic system must vary inthis manner. This is also the case for the pulmonary system so that, assumingthe time through the heart varies similarly, the circulation time TB is predictedto obey the following scaling relation

TB ∝ M1/4 (1.21)

Prosser and Brown [24] have collected measured values of circulation time forthe rabbit (8 s), the dog (16 s), and the human (23 s). These may be examinedin terms of the indicated scaling relation using the human (M = 70 kg) as refer-ence. Assuming typical body masses for the rabbit and the dog of 2 and 20 kg,


respectively, this equation provides a scaled circulation time of 9 s for the rabbitand 17 s for the dog. These findings are in good agreement with the indicatedmeasurements of 8 and 16 s, respectively.

Vascular Form and Function of the Kidney

The kidney is the organ of the body responsible for removal of waste fluid fromthe blood. The vascular organization of the kidney is of special interest becauseit provides the means for both filtering fluid from the blood and reabsorbingthe valuable portion for further use. This process is possible because of a seriesconnection of capillary networks in the kidney, one that removes fluid from theblood and the other, that allows the useful parts to be returned to it.

The basic unit of the kidney is the nephron , which consists of a capillarynetwork, referred to as the glomerulus , from which fluid is filtered, and a secondcapillary network, referred to as the peritubular capillaries , in which some ofthe fluid and valuable solutes are reabsorbed. A nephron unit is illustrated inFigure 1.15. The two kidneys of mammals contain many thousands of these units,each operating independently to produce urine, which is stored in the bladder forperiodic discharge. The operation of the units is as follows. Blood flows intothe glomerulus, and fluid is filtered through small pores in the capillary walls.The fluid is collected in a capsule and drained through a series of tubules to thepelvis of the kidney, where it then empties into the bladder. During this drainage,

Glomerulus Bowman's capsule

Peritubularcapillaries

Collectingtubule

Blood

Blood

Vasarecta

Loop of Henle

Urine to pelvis

Proximal tubule

Figure 1.15. Nephron or basic capillary unit in the kidney [1].


much of the fluid and valuable solutes are reabsorbed into the blood through theperitubular capillary vessels surrounding the tubules.

Interest here is in both form and function of the kidney for different mammalsbecause both are connected to the scaling laws for the capillary networks. Formwill involve the basic makeup of the nephron and the number present in the kidneyof a mammal. Function will involve the rate of transfer of fluid from and to thecapillaries and their difference, which equals the rate of urine production. The firstwill provide an independent confirmation of the scaling law for capillary number,and the second will indicate the scaling law for the thickness of the capillarywalls. These issues have been discussed earlier [1] and will be summarized here.

With respect to form, it may be assumed tentatively that the basic constructionpattern of the nephron is the same for all mammals and, in particular, that thenumber of capillaries in a nephron is the same for all. The product of this numberand the total number of nephrons in a kidney will thus equal the number ofcapillaries involved in the kidney function. This number can be expected to beproportional under change of scale to the total number of capillaries in the kidney,with this number proportional, in turn, to the characteristic number of capillariesNC for the entire body, as given by the third of relations (1.13). Thus, if NN

denotes the total number of nephrons in a kidney, it follows that this numbershould be proportional under change of scale to mammal mass to the power 5

8 :

NN ∝ NC ∝ M5/8 (1.22)

Interestingly, Adolph [25] applied best-fit methods of analysis to data reportedearlier by Kunkel [26] and found a power 0.62, which is essentially 5

8 (0.625).Writing the exponent as 5

8 and expressing the number of nephrons in units of1000, the scaling equation may be written as

NN = 94M5/8 (1.23)

Predictions from this equation are compared with the original countings of Kunkelin Figure 1.16, where the data can be seen to follow closely the predictions ofrelation (1.23). Thus, as to form, the scaling law for capillary number determinesthe variation in number of nephrons with mammal size; or, alternatively, thescaling law for the number of nephrons provides confirmation of the scaling lawfor the number of capillaries.

With respect to function, the matter of interest is the rate of removal of wastefluid from the blood. The capillaries of the kidneys, like those elsewhere in thebody, have numerous pores in their walls through which fluid can pass into or outof the circulating blood. The basic equation for this process resembles a diffusionprocess, except that the driving force is the net pressure �P in the capillariesarising from the blood pressure, the tissue pressure, and the osmotic pressure ofthe plasma proteins. In simplest terms, the equation for flow Qf into or out of


1

10

100

1000

10000

0.01 0.1 1 10 100 1000


Num

ber

of N

ephr

ons

(tho

usan

ds)

Mouse

Ox

Pig

Rabbit

Rat

NN = 94 M 5/8

Data of Kunkel[26]

Kidney (Nephra Number)

Figure 1.16. Scaling law for number of nephrons in a kidney and comparison with data.

a capillary network may be expressed like equation (1.18) for oxygen transfer,that is, as

Qf ∝ �PRCNCLC

HC

(1.24)

If the net pressure is assumed to be scale-invariant and the capillary wall thicknessis assumed to vary as the radius under scale change (as earlier in the case ofoxygen transfer), the following relation must apply:

Qf ∝ NCLC ∝ M5/6 (1.25)

In the glomerular capillaries, the net pressure is outward, and water and solutesthus pass through theses capillaries and into the surrounding capsule. In contrast,in the peritubular capillaries, the net pressure is inward so that some of theglomerular flow is directed back to the blood through these capillaries. In bothcases, the theoretical scaling law is that of relation (1.25). The difference in theserates is the rate of urine output QU whose scaling is thus also governed by thesame law. Predictions from this law in equation form are shown in Figure 1.17and compared with measurements summarized by Adolph [27]. The agreementcan be seen to be very good.


0.01

0.1

1

10

100

1000

10000

0.01 0.1 101 100 1000 10000


Urin

e O

utpu

t (gr

ams/

hour

)

Mouse

Elephant

Human

Rat

Dog

Horse

QU = 2.1 M 5/6

Kidney (Urine Output)

Data from Adolph [27]

Figure 1.17. Comparison of measurements of urine output of mammals with the scalinglaw of relation (1.25) in best-fit equation form.

Application to Drug Therapy

The development of the scaling relation for the time history of therapeutic drugconcentrations in small mammals and humans provides a practical applicationof the scaling laws considered here. This matter will be discussed briefly withattention restricted to the chemotherapy drug methotrexate, which is a well-known(nonmetabolized) drug used in the treatment of certain types of cancer. More than37 years ago, Dedrick et al. [28] showed that data from various mammals couldbe consolidated reasonably well into a broad correlation by plotting the ratioof drug concentration in the blood to initial dose as a function of time sinceinjection, when the latter was divided by mammal mass raised to the power 1

4 .From the previous discussion, it can now be seen that the times used in thisdescription were expressed relative to the circulation time. Observed decreasesin drug concentration in the blood with time can thus be attributed mainly todecreases associated with each cycle of blood circulation.

In considering this subject in terms of the present work, it may be anticipatedthat some refinement could be gained by including the capillary geometry andscaling into the correlation. This matter has been considered earlier by the writer[4]. For purposes of discussion and analysis, the drug is assumed to be injectedon the systemic venous side in a relatively short period of time that is negligiblein comparison with times of interest. It is also assumed that the injected drug inthe blood follows the processes of filtration, absorption and diffusion that apply


generally to the transfer of capillary fluid to and from the tissues and within thekidneys.

No information is available on the amount of the drug that is removed from theblood and then returned to it during a cycle of circulation. For scaling purposes, asimplified process may be assumed where the entire drug in the blood is removedand some of it returned during this period. In this case, the drug concentrationat the injection site after each cycle will equal the amount returned to the blood.This assumption, if not strictly correct, would lead to a calculated concentration atthe site that is less than that for the actual process. However, the same percentagedifference would be expected for all mammals, so that correct scaling would stillbe possible.

The concentration of drug (in units of mass per volume of blood) after thefirst complete cycle of blood circulation in the process simplified above cannow be considered. It has been shown earlier, in connection with scaling ofkidney function, that the flow of fluid and dissolved substances associated withthe filtering and absorption process can be considered proportional to the netcapillary length N CLC for scaling purposes. The same is also true for exchangesby diffusion. The net volume of drug reabsorbed into the blood over the time TB

for a complete cycle of circulation can be considered proportional to the productN CLCT B. The mass of drug �MD reabsorbed is proportional to this productwhen multiplied by the initial drug concentration MD0/VB. On dividing �M D

and N CLCT B by the blood volume, the expression for �M D/VB can thus bewritten as

�MD

VB

∝ MD0

VB

NCLCTB

VB

or C ∝ D0NCLCTB

VB

(1.26)

where C (=�M D/VB) denotes the concentration of the drug after the first cycleof circulation and D0(= M D0/VB) denotes the initial concentration of the drug.

With additional cycles of blood circulation, a gradual reduction and redistri-bution of the drug in the blood will occur, as a result of transfer processes inthe capillaries and net loss in the kidney, and this will further reduce the con-centration at the injection site. This process depends on the number of cycles ofcirculation and can thus be described by a factor f (t /T B), where f ( ) denotes anunspecified function. The concentration of drug in the blood is then describedfor scaling purposes by the following proportional relation:

C

D0∝ NCLCTB

VB

f

(t

TB

)(1.27)

Now, if the ratio t /T B is fixed, the ratio on the left side of this relation willbe proportional to the first ratio on the right. This provides the scaling law forthe relative concentration C /D0 as a function of time. The product N CLC varieswith mammal mass to the power 5

6 , the total blood volume VB varies directlywith mammal mass, and the time for circulation T B varies as mammal mass to


the power 14 . The desired scaling relations may be written, for example, for the

human relative to a smaller (or larger) mammal as

(C

D0

)H

=(

MH

MM

)1/12 (C

D0

)M

and tH =(

MH

MM

)1/4

tM (1.28)

where the subscripts H and M denotes values for the human and mammal,respectively.

Figure 1.18 illustrates application of this scaling law in projecting measure-ments from the mouse (mass of 0.022 kg) to the human (mass of 70 kg) for theplasma concentration of methotrexate in the blood as a function of time. Here,concentration C is expressed in micrograms per milliliter of blood plasma andinitial dose is expressed in terms of milligrams per kilogram of body weight, with3 mg/kg used for the mouse and 10 mg/kg for the human. The basic measure-ments for the mouse, as used in developing the results of Figure 1.18, are due toDedrick et al. [28], and the data for the human are due to Henderson et al. [29].The solid curve shown in the figure is a simple “best fit” power-law expressionfor measured and scaled values and is included for comparison purposes.

The agreement of the scaled data from the mouse with that of the human isindeed very good. The factor (M H/M M)1/12 in the first of the scaling relations(1.28) represents the effect of capillary process and is equal to 1.96. Without thisfactor, the predictions from the mouse would thus be reduced by a factor of ∼2.

0

1

2

3

4

0 100 200 300 400 500 600 700 800

Time (min)

Con

cent

ratio

n pe

r In

itial

Dos

e

Measurement from Human

Scaled from Mouse

Methotrexate(Chemotherapy Drug)

Figure 1.18. Graph showing drug concentration (µg/mL of blood plasma) per unit ofinitial dose (mg/kg of body weight) as a function of time for the human and as scaledfrom the mouse [4].

REFERENCES 33

Overall, the factor can be seen to improve the predictions over what they wouldbe otherwise.

1.10. EFFECTS OF STRENUOUS EXERCISE

The modeling of the vascular system discussed here is based on the assumptionof resting or near-resting conditions. For the vascular system itself, the length andradius of the vessels of mammals do not change with exercise and there can beonly one set of scaling equations for vessel dimensions. These are the relationsdeveloped and confirmed here. In contrast, the scaling relation for the numberof capillary vessels is subject to change since it is well known from the originalwork of Krogh [30] that reserve capillaries exist in the locomotive musculature,which open and allow increased oxygen delivery during strenuous exercise.

For modest exercise, the same basic scaling relations discussed here for restingphysiological functions can be expected to apply, with only the coefficients inthe scaling equations changed to accommodate the physical changes involved.However, for the extreme case of strenuous exercise, this is not the case andassociated empirical equations describing the scaling of heart rate, cardiac output,and oxygen consumption rate are known to be different from the scaling laws forthe resting state [31–33]. No detailed modeling theory is yet available for thecase of strenuous exercise, but the subject is of obvious interest in understandingthe physiological adaptations required to handle this extreme condition [34].

REFERENCES

1. Dawson TH: Engineering Design of the Cardiovascular System of Mammals ,Prentice-Hall, Englewood Cliffs, NJ, 1991.

2. Dawson TH: Similitude in the cardiovascular system of mammals, J Exp Biol204:395–407 (2001).

3. Dawson TH: Scaling laws for capillary vessels of mammals at rest and in exercise,Proc Roy Soc Lond B 270:755–763 (2003).

4. Dawson TH: Modeling of vascular networks, J Exp Biol 208:1687–1694 (2005).

5. Lambert R, Teissier G: Theorie de la similitude biologique (Theory of biologicalsimilitude), Ann Physiol 3:212–246 (1927).

6. Kenner T: Flow and pressure in the arteries, in Fung YC, Perrone N, Anliker M, eds,Biomechanics—Its Foundations and Objectives , Prentice-Hall, Englewood Cliffs, NJ,1972, pp 381–434.

7. Green HD: Circulation: Physical principles, in Glasser O, ed, Medical Physics , Year-book Publishers, Chicago, 1944, Vol 1, pp 208–232.

8. Holt JP, Rhode EA, Holt WW, Kines H: Geometric similarity of aorta, venae cavae,and certain of their branches in mammals, Am J Physiol 241:100–104 (1981).

9. Selkurt EE, Bullard RW: The heart as a pump: Mechanical correlates of cardiacactivity, in Selkurt EE, ed, Physiology , Boston; Little Brown, 1971, pp 275–295.


10. Noordergraaf A: Circulatory System Dynamics , Academic Press, New York, 1978.

11. Holt JP, Rhode EA, Kines H: Ventricular volumes and body weight in mammals, AmJ Physiol 215:704–715 (1968).

12. Brody S: Bioenergetics and Growth , Reinhold, New York, 1945.

13. Burrows TMO, Campbell IA, Howe EJ, Young, JZ: Conduction velocity and diameterof nerve fibers of cephalopods, J Physiol 179:39–40 (1965).

14. Pearson KG, Stein RB, Malhorta SK: Properties of action potentials from insect motornerves, J Exp Biol 53:299–316 (1970).

15. Clark AJ: Comparative Physiology of the Heart , Cambridge Univ Press, Cambridge,UK, 1927.

16. Marque V, Essen HV, Struijker-Boudier HAJ, Atkinson J, Lartoud-Idjouadiene I:Determination of aortic elastic modulus by pulse wave velocity and wall tracking ina rat model of aortic stiffness, J Vasc Res 38:546–550 (2001).

17. Noordergraaf A, Li JKJ, Campbell KB: Mammalian hemodynamics: A new similarityprinciple, J Theor Biol 79:485–489 (1979).

18. Gregg DE, Eckstein RW, Fineberg MH: Pressure pulses and blood pressure values inunanesthetized dogs, Am J Physiol 118:399–410 (1937).

19. Woodbury RA, Hamilton WF: Blood pressure studies in small animals, Am J Physiol119:663–674 (1937).

20. Szymanski F: Quelques solutions exactes des equations de l’hydrodynamiquede fluidevisqueux dans le cas d’un tube cylindrique (Some exact solutions to equations onhydrodynamics of viscous fluid in the case of a cylindrical tube), J Math Pures Appl11(9):67–107 (1932).

21. Gehr P, Mwangi DK, Ammann A, Malooig GMD, Taylor CR, Weibel ER: Designof the mammalian respiratory system. V. Scaling morphometric pulmonary diffusingcapacity to body mass: Wild and domestic mammals, Resp Physiol 44:61–86 (1981).

22. Schmidt-Nielsen K, Larimer JL: Oxygendissociation curves of mammalian blood inrelation to body size, Am J Physiol 195(2):424–428 (1958).

23. Schmidt-Nielsen K, Pennycuik P: Capillary density in mammals in relation to bodysize and oxygen consumption, Am J Physiol 200(4):746–750 (1961).

24. Prosser CL, Brown FA, Jr: Comparative Animal Physiology , Saunders, Philadelphia,1961.

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26. Kunkel PA Jr: The number and size of the glomeruli in the kidney of several mammals,Bull Johns Hopkins Hosp 47:285–291 (1930).

27. Adolph EF: Physiological Regulation , The Jaques Cattell Press, Lancaster, PA, 1943.

28. Dedrick RL, Bischoff KB, Zaharko DS: 1970. Interspecies correlation of plasmaconcentration history of methotrexate, Cancer Chemotherapy Report , US Natl InstHealth, Bethesda, MD, 1970, Vol 54, pp 95–101.

29. Henderson, ES, Adamson RH, Oliverio VT: The metabolic fate of tritiated methotrex-ate: Absorption and excretion in man, Cancer Res 25:1018–1024 (1965).

REFERENCES 35

30. Krogh A: A contribution to the physiology of capillaries, Nobel Lecture, Stockholm;also in Nobel Lectures, Physiology or Medicine 1901–1921 , Elsevier, Amsterdam,1967.

31. Bishop CM: Heart mass and the maximum cardiac output of birds and mammals:Implications for estimating the maximum aerobic power input of flying mammals,Phil Trans Roy Soc Lond B 352:447–456 (1997).

32. Bishop CM: The maximum oxygen consumption and aerobic scope of birds andmammals: Getting to the heart of the matter, Proc Roy Soc Lond B 266:2275–2281(1999).

33. Weibel EH, Bacigalupe LD, Schmitt B, Hoppeler H: Allometric scaling of maximalmetabolic rate in mammals: Muscle aerobic capacity as determinant factor, RespPhysiol Neurobiol 140:115–132 (2004).

34. Weibel EH, Hoppeler H: Exercise-induced maximal metabolic rate scales with muscleaerobic capacity, J Exp Biol 208:1635–1644 (2005).

Documents

CHAPTER 1 Modeling the Vascular System and Its Capillary ......4 MODELING THE VASCULAR SYSTEM AND ITS CAPILLARY NETWORKS 1.3. THE VASCULAR SYSTEM It has long been known that the vascular